Some events do not satisfy the assumptions required for a Poisson distribution, and so the use of a Poisson model leads to wrong approximations for their probabilities.
The number of students who arrive at the student union per minute will likely not follow a Poisson distribution, because the rate is not constant (low rate during class time, high rate between class times) and the arrivals of individual students are not independent (students tend to come in groups).
The number of magnitude 5 earthquakes per year in California may not follow a Poisson distribution if one large earthquake increases the probability of aftershocks of similar magnitude.
Among patients admitted to the intensive care unit of a hospital, the number of days that the patients spend in the ICU is not Poisson distributed because the number of days cannot be zero. The distribution may be modeled using a Zero-truncated Poisson distribution.
Count distributions in which the number of intervals with zero events is higher than predicted by a Poisson model may be modeled using a Zero-inflated model.